Safety Factor of Beam-Column Structures Based on Finite Element Linear Buckling Analysis

Transcription

1 Safety Factor of eam-olumn Structures ased on Finite Element Linear uckling nalysis V.N. LE a, H. HMPLIU b a. École de Technologie Supérieure, rue Notre-ame Ouest, MONTRÉL, H3 1K3 (N) b. École de Technologie Supérieure, rue Notre-ame Ouest, MONTRÉL, H3 1K3 (N) bstract : The linear buckling theory remains widely applied in structure design thanks to explicit formulas for simple structures, or robust numerical methods for solving Eigen values and Eigen vectors in linear buckling of complex structures. Linear buckling analysis by Finite Element Method (FEM) quickly gives the load multiplication factor to produce elastic buckling. This factor could be considered as the safety factor against buckling if the structure clearly remains in elastic range up to the buckling load. However, this interpretation could become unsafe if the theoretical buckling compression stress is not far enough under the yield stress. correction procedure must then be used and may be summarized into the following steps: (1) arrying out a linear static analysis with real applied loads ; (2) oing next a linear buckling analysis that uses stresses given by previous static analysis for calculating 2 nd order stiffness matrices ; This analysis would give an elastic load multiplication factor «f E» and illustrate the corresponding buckling mode; Identifying the most critical member which shows the maximum deflection in the buckling mode ; (3) Going back to static analysis results of step (1) for reading the maximum membrane stress «σ m» in the critical member and calculate the elastic buckling stress by S cre = f * σ m ; (4) If S cre is low enough, the factor f could be accepted as the safety factor against buckling without any correction, otherwise, a correction procedure must be applied to the safety factor for considering plastic deformation due to bending of column prior to buckling. In a similar philosophy of Johnson s empirical formulas for short and single columns, a practical procedure is outlined in the present work and successfully applied to three numerical examples of slender, medium and short beam-column structures for correcting the load multiplication factor given by FEM. Key words : beam-column structure, safety factor, linear buckling, load multiplication factor, finite element method, 2 nd order stiffness 1 Introduction Even if buckling of structures is highly non linear, the linear buckling theory remains widely applied in structure design thanks to explicit formulas for simple columns and robust numerical methods for solving Eigen values and Eigen vectors in linear buckling of complex structures. Linear buckling analysis by Finite Element Method (FEM) quickly gives the load multiplication factor that would produce elastic buckling. This factor could be considered as the safety factor against buckling if the membrane compressive stresses in the cross section of critical members still remain sufficiently below yield stress. Otherwise, care must be taken to correct the load multiplication factor given by elastic buckling analyses. complete theory of linear buckling with examples is given in Timoshenko [1]. This theory is widely applied in linear buckling analysis of complex structures using Finite Element Method (FEM), which can be found in many text books about FEM such as ook et al. [2]. Some references are found on application of FEM for analyzing particular and advanced buckling cases, such as Papangelis [3], Venkateswara [4] and Miyazaki [5]. but there seems to be no reference about cares to be taken when using linear buckling by FEM for structure design. Since stresses given by FEM need good interpretation and judgement for avoiding misusing them, the present work proposes a procedure of interpreting stresses of a static analysis followed by a linear elastic buckling analysis by FEM to determine the safety factor of beam-column structures. 1

2 2 Summary of Linear uckling nalysis by Finite Element Method uckling must be considered in design of every structure subjected to compression stresses over the entire cross sections such as an example of beam column structure shown in FIG. 1 and many structures with slender members or thin plates. Linear buckling analysis of structures using FEM consists of two steps summarized as in sections 2.1 and 2.2 below. 2.1 Linear Static nalysis The first step consists of building a geometric FE model, specifying boundary conditions, applying loads on the model and solving for all degrees of freedom (OF) of the model. The computation sequence in static analysis is as follows. FIG. 1 - n example of 3 beamcolumn structure Equilibrium equations of each element This stage calculates the 1 st order stiffness matrix [K 1 e ] and the applied load vector {F a e } according to matrix equation (1), where the subscript 1 designates 1 st order (using zero stress state geometry), the subscript e is for element and a for applied loads. The? mark in this equation means to show that the OFs { e } and internal forces of element {F int e } are all unknown at this stage. The number of unknowns in { e } and {F int e } is twice the number of equations, so that equations (1) could not be solved individually. [K 1 e ]{ e }? = {F a e } + {F int e }? (1) Equilibrium equations of the whole model This stage calculates the 1 st order total stiffness matrix [K 1 t ] and the total applied load vector {F a t } according to equation (2) by assembling the individual element matrices [K 1 e ] and vectors {F a e } together, where the subscript t designates total (i.e. assembly) and ext for external loads. Here, all the external forces in {F ext } are known for each free OF except the reaction forces for which the OF are known by restrained boundary condition. The total number of unknowns in {} and in {F ext } is exactly equal to the total number of OF of the model, so that (2) can be solved for all OF {} and reaction forces. [K 1 t ]{} = {F a t } + {F ext } {} = [K 1 t ] -1 {F a t + F ext } (2) Post processing calculations t this stage, all OF {} are known. If requested next, the internal forces {F int e } are calculated form equations (1) and some selected or all stresses {σ} are calculated using appropriate constitutive relationships such as (3), where [] represents the displacement-strain transformation matrix and [λ] represents the stressstrain transformation matrix, which contains elastic material properties. {σ} = [λ][]{ e } (3) 2.2 Linear uckling nalysis The linear buckling analysis, if requested, can be done after the static analysis Equilibrium equations taking the 2 nd order stiffness into account It is demonstrated that the 2 nd order stiffness is in function of the geometry and the membrane stress {σ m } (but independent of bending stress) across entire sections of each element such as shown in (4), Timoshenko [1] and ook [2]. The element 2 nd order stiffness matrices [K 2 e ] are computed based on membrane stresses given by static analysis and assembled together to form the total 2 nd order stiffness matrix [K 2 t ]. [K 2 t ] = Function of geometry and {σ m } (4) y assuming that the 2 nd order stiffness is proportional to stresses, then if the load is multiplied by a factor f, the assembly stiffness equation can be written by (5) in which, the sign convention has been changed on purpose so that [K 2 ] means stiffness reduction instead of stiffness increase. 2

3 [K 1 t - f * K 2 t ]{} = {F a t + F ext } (5) Solving Eigen value and Eigen vector equations The purpose of linear buckling analysis is to determine the multiplication factor f to bring the applied load to the buckling level at which the displacements would indefinitely increase without increasing the load: { } not all zero while { F} = {zeros}, and the equation (5) becomes an Eigenvector equations as follows. [K 1 t - f * K 2 t ]{ } = {zeros} (6) which give either { } = {zeros} for any arbitrary value of f or { } = {not zeros} if the determinant of the matrix [K 1 t - f * K 2 t ] is null, i.e. K 1 t - f * K 2 t = 0 (7) Solving the equation (7) gives several Eigen values f i, i = 1, 2,..., n. In practice, the lowest positive value, designated by f E, is called the elastic load multiplication factor for linear buckling. 3 Numerical examples of linear buckling of beam-column structures 300 kn total load 16 m 8 m (a) ase 1 Flexural stiffness of : EI = E12 N mm 4 8 m 1200 kn total load (b) ase 2 4 m 5 m 1600 kn total load (c) ase 3 2,5 m Elastic plastic material properties Young s modulus E = MPa Yield stress S Y = 250 MPa 10x290 mm 200x10 mm 200x10 mm (d) olumn cross section FIG. 2 Geometric, material and load details of three numerical examples 3.1 escription of FE model Most of details about geometry, material properties and load of three numerical examples are shown in FIG. 2. The beam is assumed to be welded on top of two columns. and are meshed into shell quadrilateral elements with mid side nodes and into 3 beam elements. The structure is assumed fixed at ends and and simply supported at top of left end in the horizontal direction. ll the loads are applied vertically and downward. 98% of the load is uniformly distributed on and 2% is applied at top of the right column. This would create slightly different mean stresses in columns and in the ratios of about 49% in and 51% in. oth static and linear buckling analyses are asked for in these examples. Three following data are the same for three cases: material properties, horizontal beam and cross section of both columns. Two following data are different for each case: the height of column and the total applied load (16 m tall with 300 kn applied load for case one, 8 m tall with 1200 kn applied load for case two and 5 m tall with 1600 kn applied load for case three). 3.2 Numerical results of static and linear buckling analyses The most relevant results of static and linear buckling analyses for three cases are summarized in Table 1, FIG. 3 and FIG. 4. These results are obtained using NSYS Finite Element ode [6]. 3

4 Table 1 Some relevant data, static analysis results and linear buckling analysis results ata : Height of column, H = Total applied load, P = ase 1 ase 2 ase 3 16 m 300 kn 8 m 1200 kn 5 m 1600 kn Maximum compression stress, σ max comp = 31,60 MPa 147,76 MPa 229,86 MPa Membrane compression in, σ m = 21,00 MPa 84,09 MPa 112,27 MPa Membrane compression in, σ m = 21,78 MPa 86,89 MPa 115,46 MPa Elastic load multiplication factor given by linear buckling analysis, f E = 2,7533 2,7391 5,1784 FIG. 3 eformed shape from static analysis (H = 16 m, P = 300 kn) FIG. 4 Mode shape from buckling analysis (H = 8 m, P = 1200 kn) 3.3 Some discussions about static and linear buckling results The case 1 is the most slender (tallest) so that it would withstand the lowest applied load. The maximum compressive axial stress σ max comp is about 1,5 times the membrane stress σ m for case one, about 1,76 times σ m for case two and about 2 times σ m for case three. These could be explained by the bending effect of horizontal beam on the columns, which is lowest for lowest applied load (case one) and highest for highest applied load (case three). The maximum axial stress (σ max comp ) may seem to be more critical than the membrane stresses. It is however wrong because the buckling of columns depends only on the membrane compression stress but not on the bending stress : the tension side of bending increases the stiffness and vanishes the reduction of stiffness due to the compression side of bending. Thus, the membrane compression stress must be considered, and stresses due to bending effect must be ignored in the interpretation of safety factor against buckling. Therefore, load multiplication factor of linear buckling analysis must be carefully checked with yield stress and must be corrected if necessary in order to safely determine the safety factor against buckling. 4 Johnson s empirical procedure for buckling of single columns uckling of single columns is well documented by several text books such as Popov [7]. The linear buckling theory of one single column gives the theoretical Euler s formula (8) for critical stress (S cre ). This stress agrees with reality for slender columns but must be corrected for short columns due to plastic deformations prior to buckling. The Johnson s empirical correction procedure is given by (9a) and (9b). 4

5 Euler s theoretical buckling stress : S cre = 2 π E I (k L) Johnson s empirical correction procedure: If S cre ½ S Y then S cr = S cre (9a) SY If S cre > ½ S Y then S cr = S Y 1-4 S where E and S Y are material Young s modulus and yield stress, I and are the cross section moment of inertia and area, L is the length of column, and k is a constant depending on restrain conditions at ends of column, for example, k = 1 for two simply supported ends, k = 0.5 for two fixed ends, etc. Some numerical examples of Johnson s procedure for slender, medium, short and very short columns are shown in Table 2. Table 2 uckling stresses of single columns according to Eulers and Johnson s formulas Euler formula : (8) Johnson formulas : (9) Material : Steel having E = MPa and S Y = 250 MPa Slender column (kl) 2 /(I/) = 200 Medium column (kl) 2 /(I/) = Short column (kl) 2 /(I/) = 60 cre (8) (9b) Very short column (kl) 2 /(I/) = 20 Euler s buckling, S cre = 49,35 MPa 137,08 MPa 548,31 MPa 4934,8 MPa Johnson s : S cr = 49,35 MPa 136,01 MPa 221,50 MPa 246,83 MPa 5 Procedure similar to Johnson s based on linear buckling by FEM y imitating Johnson s empirical correction procedure for buckling of singles columns, the following procedure is proposed for correcting the load multiplication factor given by linear buckling analysis by FEM. 5.1 Identifying the most critical member during buckling This can be done by examining the plot of the buckling mode shape. The most critical member is where the displacement is maximum. For example, the column in FIG. 4 is the most critical member even if the membrane stress in the other column () is higher in compression (see Table 1). 5.2 Reading the maximum membrane stress in critical member fter the identification of the most critical member during buckling from the mode shape, one usually needs to go back to static result files and read the maximum membrane stress (σ m ) across all sections in the critical member. ttention: the membrane stress is defined as the mean or direct stress and must not include bending stresses. 5.3 Evaluating the critical buckling stress If the load is multiplied by the elastic multiplication factor given by linear elastic buckling analysis (f E ) the membrane stress in the most critical member would become the critical stress, designated by S cre : S cre = f E* σ m (10) orrection of critical stress (S cr ) : If S cre ½ S Y then S cr = S cre (11a) SY If S cre > ½ S Y then S cr = S Y 1-4 S 5.4 Evaluating the safety factor against buckling Since the loads f E* P and f * P would give the corresponding critical stresses S cre and S cr, respectively, one must have the following proportionality: f * P / f E* P = S cr / S cre orrected safety factor f = f E* S cr /S cre (12) 5 cre (11b)

6 5.5 orrection factors for three previously studied examples y applying the procedure to the numerical results in Table 1, the elastic critical stress S cre, the corrected critical stress Scr and the corrected safety factor against buckling of three previously described cases are summarized in Table 3. Table 3 Elastic critical stress, corrected critical stress and corrected safety factor against buckling ase 1 ase 2 ase 3 ata : Height and load P 16 m & 300 kn 8 m & 1200 kn 5 m & 1600 kn Membrane compression in, σ m = 21,00 MPa 84,09 MPa 112,27 MPa Elastic load multiplication factor given by linear buckling analysis, f E = 2,7533 2,7391 5,1784 Elastic critical stress, S cre = f E* σ m 57,81 MPa 230,32 MPa 581,38 MPa orrected critical stress, S cr = 57,81 MPa 182,16 MPa 223,12 MPa orrected safety factor against buckling, f 2,7533 2, It is noticed that for the slender structure (case 1), the correction procedure does not change the linear buckling results. For short structures however, the correction may shift down significantly the load multiplication factor such as from 5.18 down to 1.99 for the case onclusions lthough finite element method is practical nowadays for design of structures, care must be given to linear buckling analysis when evaluating the safety factor of structure against buckling. The correction procedure described in section 5 of the present work constitutes a practical guide for correcting the load multiplication factor given by linear buckling analysis using finite element method with the similar assumption as Johnson s empirical correction procedure for buckling of single columns. References [1] Timoshenko S., Theory of Elastic Stability, McGraw-Hill ook ompany, [2] ook R.., Malkus. S., Plesha M. E., oncepts and pplications of Finite Element nalysis, John Wiley & Sons, 3 rd Edition, [3] Papangelis J. P., Trahair N. S., Finite Element nalysis of rch Lateral uckling, Transactions of the Institution of Engineers, ustralia, ivil Engineering, 1987, vol. 29, No. 1, pp [4] Venkateswara R. G., Kanaka R. K., Thermal Postbuckling of olumns, I Journal, 1984, vol. 22, No. 6, pp [5] Miyazaki N., reep uckling nalysis of ircular ylindrical Shells under xial ompression- ifurcation uckling nalysis by the Finite Element Method, Journal of Pressure Vessel Technology, 1987, vol. 109, No. 2, pp [6] NSYS Finite Element ode, anonsburg, US, [7] Popov E. P., Mechanics of Materials, 2 nd Edition, Prentice-Hall Inc., Englewood liffs, NJ07632,